Freezing Stochastic Travelling Waves

TL;DR

This paper investigates stochastic travelling wave solutions in PDEs, comparing methods for wave position estimation and analyzing the impact of different noise types on wave speed and behavior.

Contribution

It introduces an alternative wave position estimation method via $L^2$ norm minimization and explores its stability and effectiveness in stochastic PDEs.

Findings

Wave speed estimation methods are compared, with fixed profile minimization performing well under certain conditions.

The $L^2$ norm minimization approach can be used to freeze waves and solve related algebraic equations.

Different noise types (Ito and Stratonovich) significantly affect wave dynamics as correlation length and noise intensity vary.

Abstract

We consider in this paper travelling wave solutions to stochastic partial differential equations and corresponding wave speed. As a particular example we consider the Nagumo equation with multiplicative noise which we mainly consider in the Stratonovich sense. A standard approach to determine the position and hence speed of a wave is to compute the evolution of a level set. We compare this approach against an alternative where the wave position is found by minimizing the $L^2$ norm against a fixed profile. This approach can also be used to stop (or freeze) the wave and obtain a stochastic partial differential algebraic equation that we then discretize and solve. Although attractive as it leads to a smaller domain size it can be numerically unstable due to large convection terms. We compare numerically the different approaches for estimating the wave speed. Minimization against a fixed profile works well provided the support of the reference function is not too narrow. We then use these techniques to investigate the effect of both \Ito and Stratonovich noise on the Nagumo equation as correlation length and noise intensity increases.